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Abstract

We design an all–dielectric Lüneburg lens as an adiabatic space–variant lattice explicitly accounting for finite film thickness. We describe an all–analytical approach to compensate for the finite height of subwavelength dielectric structures in the pass–band regime. This method calculates the effective refractive index of the infinite–height lattice from effective medium theory, then embeds a medium of the same effective index into a slab waveguide of finite height and uses the waveguide dispersion diagram to calculate a new effective index. The results are compared with the conventional numerical treatment – a direct band diagram calculation, using a modified three–dimensional lattice with the superstrate and substrate included in the cell geometry. We show that the analytical results are in good agreement with the numerical ones, and the performance of the thin–film Lüneburg lens is quite different than the estimates obtained assuming infinite height.

(a) The supercell used in the DBD method for the finite height rod lattice structure. (b) Isofrequency contour of the supercell with r = 0.50a where the first band only is shown. Labels on the lines denote the corresponding normalized frequency ωa/2πc. The bold blue line corresponds to the wavelength λ = 6a used in this paper. (c) Field distribution of the waveguide slab at a particular x slice. Color shading denotes magnetic field (Hy) distribution and black contours illustrate silicon rods.

(a) Top view and side view of the thin–film subwavelength Lüneburg lens designed by EGM method for TE mode and (b) the corresponding 3D FDTD and Hamiltonian ray tracing results. (c) Top view and side view for TM mode and (d) the corresponding 3D FDTD and ray tracing results. Red circles outline the edge of Lüneburg lens, where radius R = 30a. Blue lines are the ray tracing results and color shading denotes the field [Hy for (b) and Ey for (d)] distribution, where red is positive and blue is negative.

Structure and the corresponding 3D FDTD and Hamiltonian ray tracing for the thin–film subwavelength Lüneburg lens shown in Fig. 6, but designed using the EGM method without second–order terms when estimating the effective refractive indices.

FDTD and Hamiltonian ray–tracing results of the subwavelength Lüneburg lens made of finite height silicon rods, but designed assuming infinite height. The color conventions are the same as in Figs. 6(b) and 6(d).